subject: Is 0 A Rational Number [print this page] Let us work out on is 0 a rational numberLet us work out on is 0 a rational number. If we recall what all are rational numbers, we say that the numbers which can be written in form of p/q, where we say that p and q are the integers and the value of q is not equal to zero, then the number so written is called a rational number. We know that all the series of natural numbers, whole numbers, all positive and negative integers are called rational numbers. Even all the fraction numbers and there additive inverse are calledfraction numbers. Now the question arises if 0 is a rational number or not. When we write a number 0, we observe that zero can also be written in form of 0 / 1. Here we compare thenumber 0 / 1with the rational number p / q, we come to the conclusion that p = 0 and q = 1, which is not zero. The definition of rational number says that if any number is expressed in form of p / q , then we say that the number is a rational number. Thus we come to the conclusion that 0 is a rational number.
Also we find that with zero, closure property of addition holds true, which means that id p1/q1 is a rational number 0, and p2/ q2 is another number, then surely the closure property of addition of two rational number also holds true, which means that the sum of the two rational numbers is also a rational number.
In the same way we observe that the difference of two rational numbers, from which one of the rational number is 0, is also a rational number.
If we check for the multiplication of thetwo rational numbers, we say that when ever any rational number is multiplied bya rational number 0, it gives us a result zero. So we say that the result is again a rational number, so the closure property holds true for the multiplication of the two rational numbers.
In the same way if we divide 0 by any other rational number, we get the result zero, which shows that the closure property holds true for the division of the rational numbers.
Similarly we can check other properties of rational numbers namely associative property of rational numbers , taking one of the rational number as zero and conclude that the associative property holds true for the rational numbers addition and multiplication, where we have taken one of the rational numbers as zero.
Further checking other properties of rational numbers namelycommutative property and distributive property of the rational numbers, we say that the 0 is the rational number and it satisfies all the properties of rational number.
We should remember that 0 is the additive inverse of itself, and a special quality of the rational number zero is that it does not have any multiplicative inverse. As when we write 1/0, such number does not exist.
Sowe come to the conclusion that 0 is the rational number, but it is a special rational number, different from all other rational numbers.
